積分解説1

今回はこの積分botの積分
https://twitter.com/integralsbot/status/1429257154758774785?s=21
を解説します。

${\displaystyle {\int }_0^{\pi }\log (2\cos \frac{x}{2})\log (2\sin \frac{x}{2})dx}$
$={\displaystyle {\log }^{2}2{\int }_0^{\pi }dx+\log 2{\int }_0^{\pi }(\log \cos \frac{x}{2}+\log \sin \frac{x}{2})dx+{\int }_0^{\pi }\log \cos \frac{x}{2}\log \sin \frac{x}{2}dx}$
$={\displaystyle \pi {\log }^{2}2+{4\log 2・\frac{1}{4}\frac{\partial }{\partial s}B|}_{(s,t)=(\frac{1}{2},\frac{1}{2})}+{2・\frac{1}{8}\frac{{\partial }^2}{\partial s\partial t}B|}_{(s,t)=(\frac{1}{2},\frac{1}{2})}}$

${\displaystyle \frac{\partial }{\partial s}B=\frac{\mathrm{\Gamma }(s)\mathrm{\Gamma }(t)}{\mathrm{\Gamma }(s+t)}(\psi (s)-\psi (s+t))}$
$(s,t)=(\frac{1}{2},\frac{1}{2})$に於いて${\displaystyle -2\pi \log 2}$
また、
${\displaystyle \frac{{\partial }^2}{\partial s\partial t}B=\frac{\mathrm{\Gamma }(s)\mathrm{\Gamma }(t)}{\mathrm{\Gamma }(s+t)}((\psi (s)-\psi (s+t))(\psi (t)-\psi (s+t))-{\psi }^{(1)}(s+t))}$
これは$(s,t)=(\frac{1}{2},\frac{1}{2})$で
${\displaystyle 4\pi {\log }^{2}2-\frac{{\pi }^3}{6}}$となる
よって、求める積分の値は${\displaystyle -\frac{{\pi }^3}{24}}$